Computational Social Choice: Autumn 2013 Ulle Endriss Institute for - PowerPoint PPT Presentation

Introduction COMSOC 2013 Computational Social Choice: Autumn 2013 Ulle Endriss Institute for Logic, Language and Computation University of Amsterdam Ulle Endriss 1

Introduction COMSOC 2013 Social Choice Theory SCT studies collective decision making: how should we aggregate the preferences of the members of a group to obtain a “social preference”? Agent 1: △ ≻ � ≻ � Agent 2: � ≻ � ≻ △ Agent 3: � ≻ △ ≻ � Agent 4: � ≻ △ ≻ � Agent 5: � ≻ � ≻ △ ? SCT is traditionally studied in Economics and Political Science, but now also by “us”: Computational Social Choice . Ulle Endriss 2

Introduction COMSOC 2013 Organisational Matters Prerequisites: This is an advanced course: I assume mathematical maturity, we’ll move fast, and we’ll often touch upon recent research. On the other hand, little specific background is required (just a bit of complexity theory ). Examiniation: Homework (best n − 1 of n , 80%) + presentation (20%). Website: Lecture slides, homework assignments, papers to present, and other important information will be posted on the course website: http://www.illc.uva.nl/ ∼ ulle/teaching/comsoc/2013/ Seminars: There are occasional talks at the ILLC that are relevant to the course and that you are welcome to attend (e.g., at the COMSOC Seminar). Ulle Endriss 3

Introduction COMSOC 2013 Plan for Today Today’s lecture has two parts: • Part I. Informal introduction to some of the topics of the course • Part II. A classical result: Arrow’s Theorem Ulle Endriss 4

Introduction COMSOC 2013 Part I: Examples, Problems, Ideas Ulle Endriss 5

Introduction COMSOC 2013 Three Voting Rules How should n voters choose from a set of m alternatives ? Here are three voting rules (there are many more): • Plurality: elect the alternative ranked first most often (i.e., each voter assigns 1 point to an alternative of her choice, and the alternative receiving the most points wins) • Plurality with runoff : run a plurality election and retain the two front-runners; then run a majority contest between them • Borda: each voter gives m − 1 points to the alternative she ranks first, m − 2 to the alternative she ranks second, etc.; and the alternative with the most points wins Ulle Endriss 6

Introduction COMSOC 2013 Example: Choosing a Beverage for Lunch Consider this election with nine voters having to choose from three alternatives (namely what beverage to order for a common lunch): Milk ≻ Beer ≻ Wine 4 Dutchmen: Wine ≻ Beer ≻ Milk 3 Frenchmen: Beer ≻ Wine ≻ Milk 2 Germans: Which beverage wins the election for • the plurality rule? • plurality with runoff? • the Borda rule? Ulle Endriss 7

Introduction COMSOC 2013 Example: Electing a President Remember Florida 2000 (simplified): Bush ≻ Gore ≻ Nader 49%: Gore ≻ Nader ≻ Bush 20%: Gore ≻ Bush ≻ Nader 20%: 11%: Nader ≻ Gore ≻ Bush Questions: • Who wins? • Is that a fair outcome? • What would your advice to the Nader-supporters have been? Ulle Endriss 8

Introduction COMSOC 2013 Example: Voting in Multi-issue Elections Suppose 13 voters are asked to each vote yes or no on three issues; and we use the plurality rule for each issue independently: • 3 voters each vote for YNN, NYN, NNY. • 1 voter each votes for YYY, YYN, YNY, NYY. • No voter votes for NNN. But then NNN wins: 7 out of 13 vote no on each issue ( paradox! ). What to do instead? The number of (combinatorial) alternatives is exponential in the number of issues (e.g., 2 3 = 8 ), so even just representing the voters’ preferences is a challenge . . . S.J. Brams, D.M. Kilgour, and W.S. Zwicker. The Paradox of Multiple Elections. Social Choice and Welfare , 15(2):211–236, 1998. Ulle Endriss 9

Introduction COMSOC 2013 Judgment Aggregation Preferences are not the only structures we may wish to aggregate. In JA we aggregate people’s judgments regarding complex propositions. p → q p q Judge 1: Yes Yes Yes Judge 2: Yes No No Judge 3: No Yes No ? Ulle Endriss 10

Introduction COMSOC 2013 Fair Division Fair division is the problem of dividing one or several goods amongst two or more agents in a way that satisfies a suitable fairness criterion. One instance of this problem is cake cutting . For two agents , we can use the cut-and-choose procedure: ◮ One agent cuts the cake in two pieces (she considers to be of equal value), and the other chooses one of them (the piece she prefers). The cut-and-choose procedure is proportional: ◮ Each agent is guaranteed at least one half (general: 1 /n ) according to her own valuation. What if there are more than two agents? Is proportionality the best way of measuring fairness? What about other types of goods? Ulle Endriss 11

Introduction COMSOC 2013 Computational Social Choice Research can be broadly classified along two dimensions — The kind of social choice problem studied, e.g.: • electing a winner given individual preferences over candidates • aggregating individual judgements into a collective verdict • fairly dividing a cake given individual tastes • finding a stable matching of students to schools The kind of computational technique employed, e.g.: • algorithm design to implement complex mechanisms • complexity theory to understand limitations • logical modelling to fully formalise intuitions • knowledge representation techniques to compactly model problems • deployment in a multiagent system Y. Chevaleyre, U. Endriss, J. Lang, and N. Maudet. A Short Introduction to Computational Social Choice. Proc. SOFSEM-2007. Ulle Endriss 12

Introduction COMSOC 2013 Part II: Arrow’s Theorem Ulle Endriss 13

Introduction COMSOC 2013 Arrow’s Theorem This is probably the most famous theorem in social choice theory. It was first proved by Kenneth J. Arrow in his 1951 PhD thesis. He later received the Nobel Prize in Economic Sciences in 1972. What we will see next: • formal framework: social welfare functions • the axiomatic method in SCT, and some axioms • the theorem, its interpretation, and a proof K.J. Arrow. Social Choice and Individual Values . John Wiley and Sons, 2nd edition, 1963. First edition published in 1951. Ulle Endriss 14

Introduction COMSOC 2013 Formal Framework Basic terminology and notation: • finite set of individuals N = { 1 , . . . , n } , with n � 2 • (usually finite) set of alternatives X = { x 1 , x 2 , x 3 , . . . } • Denote the set of linear orders on X by L ( X ) . Preferences (or ballots ) are taken to be elements of L ( X ) . • A profile R = ( R 1 , . . . , R n ) ∈ L ( X ) n is a vector of preferences. • We shall write N R x ≻ y for the set of individuals that rank alternative x above alternative y under profile R . For today we are interested in preference aggregation mechanisms that map any profile of preferences to a single collective preference. The proper technical term is social welfare function (SWF): F : L ( X ) n → L ( X ) Ulle Endriss 15

Introduction COMSOC 2013 The Axiomatic Method Many important classical results in social choice theory are axiomatic . They formalise desirable properties as “ axioms ” and then establish: • Characterisation Theorems , showing that a particular (class of) mechanism(s) is the only one satisfying a given set of axioms • Impossibility Theorems , showing that there exists no aggregation mechanism satisfying a given set of axioms Ulle Endriss 16

Introduction COMSOC 2013 Anonymity and Neutrality Two very basic axioms (that we won’t actually need for the theorem): • A SWF F is anonymous if individuals are treated symmetrically: F ( R 1 , . . . , R n ) = F ( R π (1) , . . . , R π ( n ) ) for any profile R and any permutation π : N → N • A SWF F is neutral if alternatives are treated symmetrically: F ( π ( R )) = π ( F ( R )) for any profile R and any permutation π : X → X (with π extended to preferences and profiles in the natural manner) Keep in mind: • not every SWF will satisfy every axiom we state here • axioms are meant to be desirable properties (always arguable) Ulle Endriss 17

Introduction COMSOC 2013 The Pareto Condition A SWF F satisfies the Pareto condition if, whenever all individuals rank x above y , then so does society: N R x ≻ y = N implies ( x, y ) ∈ F ( R ) This is a standard condition going back to the work of the Italian economist Vilfredo Pareto (1848–1923). Ulle Endriss 18

Introduction COMSOC 2013 Independence of Irrelevant Alternatives (IIA) A SWF F satisfies IIA if the relative social ranking of two alternatives only depends on their relative individual rankings: x ≻ y = N R ′ N R x ≻ y implies ( x, y ) ∈ F ( R ) ⇔ ( x, y ) ∈ F ( R ′ ) In other words: if x is socially preferred to y , then this should not change when an individual changes her ranking of z . IIA was proposed by Arrow. Ulle Endriss 19

Introduction COMSOC 2013 Universal Domain This “axiom” is not really an axiom . . . Sometimes the fact that any SWF must be defined over all profiles is stated explicitly as a universal domain axiom. Instead, I prefer to think of this as an integral part of the definition of the framework (for now) or as a domain condition (later on). Ulle Endriss 20